Dr. Joel Friedman’s work featured in Quanta Magazine’s 2025 Biggest Breakthroughs in Mathematics

UBC Computer Science Professor Joel Friedman’s proof on graphs helped mathematicians solve a major problem in geometry. 

UBC Computer Science Professor Joel Friedman was recently featured in Quanta Magazine’s 2025 Biggest Breakthroughs in Mathematics. Dr. Friedman’s proof about spectral gap in random graphs helped two mathematicians solve a difficult problem in the hyperbolic geometry field. 

In 2002, Dr. Friedman showed that most random graphs have the largest possible spectral gap, meaning that the graph is well-connected.  

More than a decade later, mathematicians Laura Monk and Nalini Anantharaman were continuing Maryam Mirzakhani’s work on hyperbolic surfaces when they got stuck.  

But in Dr. Friedman’s paper, Dr. Monk and Dr. Anantharaman found a key formula that they adapted to calculate the average spectral gap on all hyperbolic surfaces. They showed that almost all hyperbolic surfaces are as connected as possible. This breakthrough has implications in the field of quantum chaos, a field of physics that focuses on chaos theory in quantum systems. 

"Although my proof of Alon's Second Eigenvalue conjecture is often quoted, very few people have gone through the details of the proof,” says Dr. Friedman. “I was thrilled that one very specialized detail — generalized Mobius functions — was useful to Drs. Anantharaman and Monk some two decades after my work.”